P O L Y A K O V L O O P C O R R E L A T I O N S I N L A N D A U G A U G E A N D T H E H E A V Y Q U A R K P O T E N T I A L
N. A T T I G a F. K A R S C H b, B. P E T E R S S O N a, H. SATZ a,c and M. W O L F F "
a Fakultiit fiir Physik, Universitdt Bielefeld. D-4800 Bielefeld 1, Fed. Rep. Germany b TheoryDivision, CERN, CH-1211 Geneva 23. Switzerland
c Physics Department, Brookhaven National Laboratory, Upton, N Y 11973, USA Received 1 March 1988; revised manuscript received 16 May 1988
We calculate Polyakov loop correlation functions in SU (3) gauge theory on a 123 × 4 lattice. We determine from this the colour averaged heavy quark potential and compare it with the corresponding colour singlet potential in Landau gauge. A comparison with finite temperature perturbation theory shows that perturbative relations are at best recovered for very high temperatures.
Numerical studies o f the Q C D plasma phase have shown that quantities that are sensitive to short dis- tance (large m o m e n t u m ) structure o f the plasma are well described by finite temperature perturbation theory. In particular the energy density o f a q u a r k - gluon plasma closely resembles ideal gas behaviour above the deconf'mement/chiral phase transition ~1.
Recent studies o f long distance features like Debye screening o f external colour charges [ 2,3 ] have, how- ever, shown that even at rather high temperatures numerical results do not agree with perturbative re- suits [4 ]. This may not be too surprising, as various non-perturbative modes m a y play a role in the long distance (low m o m e n t u m ) sector o f the Q C D plasma [ 1,5,6 ]. In fact it has been argued that the Debye mass itself cannot be defined perturbatively [ 7 ] and that the standard definition as the zero m o m e n t u m limit o f the zeroth c o m p o n e n t o f the v a c u u m polarization tensor is meaningless. In any case this is a gauge in- variant concept only in leading order perturbation theory.
Understanding Debye screening in non-abelian gauge theories is in itself o f fundamental importance and plays a keyrole in the discussion o f convergence o f finite temperature perturbation theory [ 8 ]. I n or- der to see whether contact can be m a d e between non- perturbative calculations o f the heavy quark poten-
~ For a review see ref. [ 1 ].
tial and finite temperature perturbation theory we studied the colour averaged heavy quark potential as well as colour singlet and octet potentials in L a n d a u gauge. We prefer this gauge over e.g. the axial gauge at finite temperature because o f the rotational sym- metry o f the gauge condition. This allows us to check various relations a m o n g these potentials given in perturbation theory [4 ].
For temperatures above deconfinement the colour averaged heavy quark potential is defined in terms o f Polyakov loop correlation functions on a lattice o f size N ~ X N ~ ,
exp [ - V ( r ) / T ] = ( T r L ( R ) Tr L * ( 0 ) ) / ( L ) 2, ( 1 ) with T r L ( R ) = T r ~ 2 1 U ( R , i ) , o denoting the Polyakov loop a n d ( L ) = ( T r L ( 0 ) ) . Temperature T and distance r are measured in units o f the lattice spacing a and are given by
1 / T = N ~ a , r = R a , R = l , 2 , . . . , N ~ / 2 . (2) At high temperature, perturbation theory predicts for this colour averaged potential
V ( r ) / T = - ~ V I ( r ) 2 / T 2 , (3)
where 111 (r) is the singlet term, which is expected to be o f Debye screened C o u l o m b f o r m
VI ( r) = [ a ( T ) / r ] e x p [ - # o ( T ) r ] , (4a)
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. 65
Volume 209, number 1 PHYSICS LETTERS B 28 July 1988
with
o~( T) = - g 2 ( T) ( N 2 - 1 ) / 8 g N . (4b) Here/ZD(T) = x / ~ g ( T ) Fis the Debye mass in the case of a pure S U ( N ) gauge theory. It should be stressed, however, that the Debye screening mass in the potential is not a perturbative result, but rather is based on the usual attempt to reproduce non-pertur- bative features in perturbative calculations by sum- ming up certain infinite subsets of diagrams [ 9 ]. Thus we are actually discussing the incorporation of non- perturbative effects in the framework o f perturbation theory when we compare eqs. (3) and (4) with Monte Carlo simulations. The simulations [2,3] so far seem to indicate that the colour averaged poten- tial behaves like V ( r ) ~ e x p ( - c r ) / r rather than showing the expected 1 / r 2 behaviour.
In order to analyze this discrepancy further and see in which sense the perturbative limit can be reached, we have studied the potential in more detail for the pure SU (3) gauge theory. In addition to the colour averaged potential defined by eq. ( I ) we also study the singlet and octet potentials defined as [ 4,10 ] e x p [ - V~ ( r ) / T ] = 3 ( T r L ( R ) L * ( 0 ) ) / ( L )
2
(5) e x p [ - V s ( r ) / T ] = 9 ( T r L ( R ) Tr L * ( 0 ) ) / ( L ) 2
- ] ( T r L ( R ) L * ( 0 ) ) / ( L ) 2
=9 e x p [ - V ( r ) / T ] - ~ e x p [ - V l ( r ) / T ] . (6) As these quantities are not gauge invariant we study them in Landau gauge. In perturbation theory the singlet and octet potentials are related by
V1 ( r ) / V s ( r ) = - 8 + O ( g 4 ) . (7a) We note that in this relation the O ( g 2 ) corrections cancel, while this is not the case for similar relations deduced from eq. (3):
Vl ( r ) 2 / T V ( r ) = - 1 6 + O ( g 2) . (7b) To study these potentials we performed Monte Carlo simulations on a 123)<4 lattice at three cou- plings fl= 5.75, 6.10 and 8.00. As the deconfinement phase transition on this sized lattices occurs at flc-~5.69 [11], the above fl values correspond to
T / T c = 1.16, 2.3 and 13.7 t,2. In units of the decon- finement temperature To. At each fl value we have performed 42 000 iterations. Measurements have been performed every tenth iteration. On these con- figurations we also fixed the Landau gauge following the procedure outlined in ref. [ 13 ], i.e. we first fixed the axial gauge and then iteratively fixed the Landau gauge by maximizing the quantity
3
R e T r Z (Ux,u+U~-,,u) (8)
a=o
on each site. The iterative process has been per- formed 100 times. We checked rotational symmetry by comparing ( T r Ux.u) in the four directions. After 100 iterations the differences between these values were found to be less than 2%.
The error analysis has been performed by subdi- viding the total data sample in 8 blocks of 512 mea- surements each. On each of these blocks we determined the potentials according to eqs. ( 1 ), (5) and (6). Errors are then calulated as statistical errors of these eight measurements of the potentials. In fig.
1 we show results for the colour averaged (fig. l a ) and singlet and octet (fig. l b ) potentials at the three fl values studied.
Our results are summerized in table 1 and shown in fig. 1. From this we see that indeed the octet poten- tial is repulsive, while the singlet potential is attrac- tive. In fig, 2 we show the ratio between singlel and octet potentials (fig. 2a) as well as - V~ ( r ) 2 / V ( r ) T (fig. 2b) as functions of R. We note that for lower temperatures the ratio - V~ ( r ) / V s ( r ) rises fast, in- dicating that the octet potential drops fast. At large distances the colour singlet potential thus gives the dominant contribution to the colour averaged poten- tial for temperatures close to To. In the perturbative regime we would expect the relations (7a) and (7b) to hold. For the largest fl value we indeed find the expected ratio between singlet and octet potentials, while the relation between colour averaged and sin- glet potential is in general not yet fulfilled. This may reflect the different order in g2 neglected in (7a) re- spectively (7b). Apparently we find the best agree- ment with perturbation theory at short distances. At large distances, the "non-perturbative" screening
~2 Here we have taken into account the observed scaling viola- tions for Tc/AL. For further details see re£ [ 12 ].
66
I - -
i
10-1
10-2
10-3
\ ]
5
10 0
~_ I0 - I
i
10-2
10-3
o •
z
I
10 -4 i , i
1 2 3 t, 6 1 2 3 ¢ 5 6
R R
Fig. 1. (a) The function - V(R) / Tversus lattice distance R on a 123 X 4 lattice at/?= 5.75 (circles), 6.10 (squares) and 8.00 (triangles), where V(R ) is the colour averaged potential. (b) The same as j n (a), with filled symbols for the singlet potential - 111 ( R ) / T and open symbols for the octet potential Vs (R) / T.
Table 1
Monte Carlo data for the colour averaged potential V(R), singlet potential V~ (R) and octet potential Vs(R) in units of the temperature T at three fl values fl= 5.75, 6.10 and 8.00. Details about the data sample and error analysis are discussed in the text.
fl ( L ) R V ( R ) / T V~(R)/T Vs(R)/T
5.75 0.53753 (191) 1 0.2790 (26) 1.6946 (63) 0.2152 (13)
2 0.0687 (17) 0.7847 (54) 0.0715 (11)
3 0.0231 (9) 0.3199 (28) 0.0211 (10)
4 0.0096 (9) 0.1138 (30) 0.0043 (8)
5 0.0051 (11) 0.0403 (28) -0.0005 (11)
6 0.0024 (11) 0.0194 (25) 0.0002 (10)
6.10 0.81948 (98) 1 0.10422 (48) 1.1156 (25) 0.1426 (5)
2 0.02005 (31) 0.4596 (20) 0.0515 (5)
3 0.00599 (14) 0.1766 (12) 0.0176 (3)
4 0.00209 (27) 0.0642 (13) 0.0060 (3)
5 0.00041 (19) 0.0222 (14) 0.0023 (3)
6 0.00066 (46) 0.0127 (24) 0.0009 (5)
8.00 1.44292 (119) l 0.02148 (14) 0.5445 (17) 0.0683 (3)
2 0.00402 (9) 0.2210 (18) 0.0267 (3)
3 0.00110 (7) 0.0969 (19) 0.0116 (2)
4 0.00032 (6) 0.0448 (18) 0.0054 (2)
5 0.00012 (6) 0.0232 (16) 0.0028 (2)
Volume 209, number 1 PHYSICS LETTERS B 28 July 1988 20
10
dk
A , .
lomb" factor and assume a simple exponential decay governed by a screening mass #. Note that this pa- rametrization gives an effective screening mass de- scribing the behaviour of the potential at intermediate distances. To study the asymptotic behaviour at large distances one could also use an ansatz including higher excited states, i.e. additional exponentials in eq. (9). We prefer here to define an effective screen- ing mass #R.d at distance R through
20
1 2 3 /.
R
I,,--
>
10
>
i
o 1 2 3 /.
R
Fig. 2. The ratios - V~ ( R ) / V s ( R ) (a) and - V~ ( R ) 2 / V ( R ) T (b) versus R for three values o f t . Symbols are as in fig. 1. The dashed line corresponds to the perturbative result.
factor exp [ --/~D (T) r ] becomes important, and we see in particular in fig. 2b that even at high temperature relation (3) is not well satisfied. On the other hand, the improved agreement at short distances shows that the "truly" perturbative aspects are better and better reproduced as T increases.
Our present analysis of the heavy quark potential is limited to distances 0.25~<rT~< 1.5, whereas the simple Debye screening form of the potential can be- come valid only in the limit r T ~ ~ . At intermediate distances the structure of the potential may be more complicated [ 14 ]. To study the functional behaviour in more detail we parametrize the potentials in the general form
V( r) / T= [ c( T) / ( rT) a] e x p ( - ~ t r ) . (9) We thus allow for an arbitrary power d in the "Cou-
V ( R ) / V(R + 1 ) = {R -d exp ( -- ~tR.aR) + ( 1 2 - R ) - d e x p [ -- #Ra(12--R) ] }
× { ( R + 1 ) - d e x p [ --#R.d(R+ 1 ) ]
+ (11 - - R ) - d exp [ --#R,a( 11 --R) ]}-1 (10) Here we take into account the periodicity of our 123×4 lattice. For any value of d these effective masses approach the same limiting value
~t= lim #R,a. (11)
R ~ o c
I f the potential drops according to a power law 1/r d°, the approximants ItR.d approach # from above (below) for d < do ( d > do ). The "best" choice for the exponent d obviously is obtained, if #R,~ is roughly R independent already at short distances. In fig. 3 we show/tR,d for the colour averaged potential and three different values of d. We see that apparently we need 1 ~< d~< 2 to get #R.d approximately R independent. In fig. 4 we show the behaviour of/tR,a for the singlet potential. Here we seem to need d ~ 0,3. Detailed re- suits for d and #, obtained from a two-parameter fit of the ratios V ( R - 1 ) / V ( R ) , are given in table 2. Let us compare these results in some detail with pertur- bative calculations.
We noticed already in connection with fig. 2 that at r = 5.75 and 6.10 we do not see any indications for perturbative behaviour. This also follows from the analysis of the functional form of the potentials. At r = 8.00 the powerlike behaviour of the singlet poten- tial is best described by d = 0 . 2 4 in the range
~3 Note that for d - 0 the effective masses rise with increasing R.
Similar behaviour has been found for the gluon propagator in Landau gauge [ 15 ]. There the increase of the effective mass has been taken as evidence for the non-positivity of the trans- fer matrix in this gauge.
68
d= 0
-1.5
-1.0
I
I 2 3
R
(a) d= 1 (b) 1.0 d= 2 (c)
-1.0 ~ 0.5
/
-o.s /
4
I
I 2 3 I 2 3
R
Fig. 3. Effective masses/zR.a in inverse lattice spacings for the colour averaged potential versus R and d = 0 (a), d = 1 (b) and d = 2 (c).
Results are shown for the three values o f fl (symbols as in fig. 1 ). Lines are drawn to guide the eye.
d= -1 (a)
:::k 2 ~
1.0
1 2 3 4.
R
-1.0
-0.5 I
I 2 3 t+ I 2 3 t~
R R
Fig. 4. The same as fig. 3, now for the colour singlet potential and d = - 1 (a), d = 0 (b) a n d d = 1 (c).
Table 2
The power d and Debye m a s s # for the colour averaged a n d sin- glet potentials obtained from a Z 2 fit of the ratios V ( R - 1 ) / V ( R ) as discussed in the text.
B Singlet Colour averaged
/t d /~ d
5.75 [.11 (1) - 0 . 4 9 (1) 0.61 (7) 1.15 (10) 6.10 t.07 (1) - 0 . 2 6 (1) 0.55 (7) 1.59 (11) 8.00 0.74 (2) 0.24 (3) 0.58 (6) 1.59 (9)
0.25 < r T < 1.5. Besides this we see from figs. 4b and 4c that the effective masses extracted for d = 0 and d = 1 seem to give quite accurate upper and lower bounds for the asymptotic value of the screening mass at fl= 8.00:
2.4< # / T < 3.2. (12)
This is to be compared with the perturbative result, /tD / T = g ( T). The large value found for # / T would thus require a rather large value for the temperature-
Volume 209, number 1 PHYSICS LETTERS B 28Ju~ 1988 d e p e n d e n t r u n n i n g coupling constant. To explain it
by lowest order p e r t u r b a t i o n theory would d e m a n d a n u n r e a s o n a b l y high A value i n the r u n n i n g coupling c o n s t a n t
g 2 ( T ) = 2 4 n 2 / 3 3 l n ( T / A ) , (13)
n a m e l y A ~> 3 T~. This i n t u r n makes it likely to expect higher order terms to be relevant i n the p e r t u r b a t i v e analysis. U n l i k e the singlet potential, which could be m e a s u r e d up to distances rT= 1.5, the colour aver- aged potential could be d e t e r m i n e d by us only up to
rT= 1. This makes the b o u n d s o n the asymptotic val- ues of the screening mass less stringent. At ,6= 8.00 we find
2.0~<#/T~<4.0. ( 1 4 )
This makes it difficult to judge whether asymptoti- cally the r e l a t i o n / t .. . . age = 2#singlet holds. To investi- gate the effects at finite lattice spacing, we have fitted o u r data to a lattice v e r s i o n o f eq. ( 9 ) , d e f i n e d from a discretized F o u r i e r transform. It t u r n s out, how- ever, that the mass value o b t a i n e d f r o m this fit dif- fers very little from the one o b t a i n e d by using eq. ( 9 ) a n d i n c l u d i n g the lattice periodicity.
At finite distances rT>~ 1 we start feeling the finite size of the lattice used at present a n d we have to worry a b o u t the influence o f these finite size effects. Work i n this direction is i n progress [ 16 ].
I n conclusion, we find that the heavy q u a r k poten- tial exhibits a complicated structure at short a n d in- termediate distances studied by us, i.e. 0.25 ~< rT<~ 1.5.
We see that for t e m p e r a t u r e s close to T¢, 1.0< T~
Te< 3.0, p e r t u r b a t i v e relations fail to describe the potential a n d no indications are f o u n d that they could describe the large distance b e h a v i o u r o f the poten-
tial. At large temperatures, T ~ t0T¢, we f i n d some i n d i c a t i o n s for the validity of p e r t u r b a t i v e relations like eqs. ( 3 ) a n d ( 7 ) . However, even here the screen- ing masses t u r n out to be rather large. P r e s u m a b l y even higher temperatures are n e e d e d to find com- plete agreement with p e r t u r b a t i o n theory.
References
[1 ] F. Karsch, Proc. Quark matter '87, Z. Phys. C, to be published.
[2] T.A. De Grand and C.E. De Tar, Phys. Rev. D 34 (1986) 2469.
[3] K. Kanaya andH. Satz0 Phys. Rev. D 34 (1986) 3193.
[4] S. Nadkarni, Phys. Rev. D 33 (1986) 3738.
[5] E. Manousakis and J. Polonyi, Phys. Rev. Lett. 58 (1987) 847.
[ 6 ] J.B. Kogut and C.A. De Tar, Phys. Rev. D 36 ( 1987 ) 2828.
[7] S. Nadkarni, Phys. Rev. D 34 (1986) 3904.
[ 8 ] A. Linde, Rep. Prog. Phys. 42 ( 1979 ) 389.
[ 9 ] See e.g., D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod.
Phys. 53 (1981) 43.
[ 10] L.S. Brown and W.I. Weisberger, Phys. Rev. D 20 (1979 ) 3239.
[11 ] S.A. Gottlieb et al., Phys. Rev. Lett. 55 (1985) 1958;
N.H. Christ and A.E. Terrano, Phys. Rev. Lett. 56 (1986) 111.
[12] N. Attig, F. Karsch, B. Petersson, H. Satz and M. Wolff, in preparation.
[ 13 ] C.T.H. Davies. Proc. Lattice gauge theory '86, eds. H. Satz, I. Harrity and J. Potvin, NATO ASI Series, Vol. B 159 (Plenum, New York).
[ 14 ] J.C. Gale and J. Kapusta, University of Minnesota preprint (July 1987).
[ 15] J.E. Mandula and M. Ogilvie, Phys. Lett. B 185 (1987) 127;
Brookhaven preprint, BNL-40440 (October 1987).
[ 16 ] J. Engels, F. Karsch and H. Satz, in preparation;
N. Attig, J. Chakrabarti, B. Petersson, H. Satz and M. Wolff, in preparation.
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